Testing for a disease can be made more efficient by combining samples. If the samples from six people positive sample will always test positive, no matter how many negative samples it is mixed with. Assum samples combined into one mixture. Is the probability low enough so that further testing of the individua

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Pooling samples—combining multiple individual samples into a single mixture for testing—is an effective strategy to increase testing efficiency, especially when the prevalence of the disease is low. This method, often referred to as **group testing** or **pooled testing**, reduces the number of tests needed by initially testing multiple samples together. If the pooled test is negative, all individuals in the pool are presumed negative. If the pooled test is positive, each sample within the pool must be tested individually to identify the positive cases.

Let's delve into the specifics based on your scenario:

### **Scenario Overview**

- **Pool Size:** 6 samples combined into one mixture.
- **Testing Rule:** If at least one sample in the pool is positive, the pooled test returns positive. Otherwise, it returns negative.

### **Efficiency Analysis**

To determine whether pooling is efficient, we need to evaluate the expected number of tests required per individual and compare it to individual testing without pooling.

1. **Probability Calculations:**
   
   - **Let $$ p $$** be the probability that an individual sample is positive.
   - **Probability the pooled test is negative:** All six samples must be negative.
     $$
     P(\text{Pool Negative}) = (1 - p)^6
     $$
   - **Probability the pooled test is positive:** At least one sample is positive.
     $$
     P(\text{Pool Positive}) = 1 - (1 - p)^6
     $$

2. **Expected Number of Tests per Pool:**
   
   - **Initial pooled test:** 1 test.
   - **If pool is positive:** Additional 6 individual tests.
   - **Total expected tests per pool:**
     $$
     \text{Expected Tests} = 1 + 6 \times P(\text{Pool Positive}) = 1 + 6 \times \left[1 - (1 - p)^6\right]
     $$
   
   - **Expected tests per individual:**
     $$
     \text{Expected Tests per Individual} = \frac{1 + 6 \times \left[1 - (1 - p)^6\right]}{6} = \frac{1}{6} + \left[1 - (1 - p)^6\right]
     $$

3. **Comparison with Individual Testing:**
   
   - **Individual testing requires 1 test per person.**
   - **Pooling is more efficient if:**
     $$
     \frac{1}{6} + \left[1 - (1 - p)^6\right] < 1
     $$
     Simplifying:
     $$
     1 - (1 - p)^6 < \frac{5}{6}
     $$
     Which is always true since $$ (1 - p)^6 > \frac{1}{6} $$ for all $$ p < 1 $$.

- **However, to determine the efficiency gain**, we look at when the expected tests per individual are significantly less than 1. This largely depends on the prevalence $$ p $$:
     
     - **Low Prevalence ($$ p $$ is small):** 
       - $$ (1 - p)^6 \approx 1 - 6p $$ (using the binomial approximation for small $$ p $$).
       - $$ \text{Expected Tests per Individual} \approx \frac{1}{6} + 6p $$
       - Pooling is efficient if $$ \frac{1}{6} + 6p < 1 $$, which holds true for $$ p < \frac{5}{36} $$ (~13.89%).

- **Higher Prevalence:** 
       - As $$ p $$ increases, $$ 1 - (1 - p)^6 $$ approaches 1, making the expected tests per individual approach $$ \frac{1}{6} + 1 = \frac{7}{6} $$, which is less efficient than individual testing.

### **Practical Implications**

- **Optimal Pool Size Depends on Prevalence:**
  - **Lower Pool Sizes (e.g., 2-4):** More efficient at higher prevalence rates compared to larger pools.
  - **Larger Pool Sizes (e.g., 6):** Best suited for very low prevalence rates.

- **Decision on Further Testing:**
  - With pooling, **further individual testing is only required if the pooled test is positive**.
  - The efficiency gain is maximized when the probability of the pooled test being negative is high, minimizing the need for additional tests.

### **Conclusion**

Pooling samples into groups of six can significantly enhance testing efficiency **provided that the prevalence of the disease is sufficiently low** (ideally below approximately 13.89%). In such cases, the majority of pooled tests will be negative, eliminating the need for individual testing and thereby conserving resources. However, as disease prevalence increases, the likelihood of pooled tests returning positive rises, diminishing the efficiency benefits and potentially resulting in more tests than individual testing would require.

**Recommendation:** Assess the current prevalence rate of the disease within the population you're testing. If it remains low, pooling in groups of six is advantageous. If prevalence increases, consider reducing the pool size or reverting to individual testing to maintain efficiency.
Pooling samples into groups of six can be efficient when the disease prevalence is low, reducing the number of tests needed. However, if the disease is more common, individual testing may be more effective.

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